Structural adaptive deconvolution under $${\mathbb{L}_p}$$ -losses

Rebelles, Gilles

doi:10.3103/s1066530716010026

Cited by 13 publications

(18 citation statements)

References 41 publications

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“…Below we will discuss only the results of Theorems 1 and 2. An estimator attaining simultaneously the asymptotics proved in Theorem 3 was recently constructed in Rebelles (2015). The latter implies that our results concerning the estimation under sup-norm loss are sharp.…”

Section: Are the Lower Bounds Sharp?mentioning

confidence: 85%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

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Abstract:The aim of the paper is to establish asymptotic lower bounds for the minimax risk in two generalized forms of the density deconvolution problem. The observation consists of an independent and identically distributed (i.i.d.) sample of n random vectors in R d . Their common probability distribution function p can be written aswhere f is the unknown function to be estimated, g is a known function, α is a known proportion, and ⋆ denotes the convolution product. The bounds on the risk are established in a very general minimax setting and for moderately ill posed convolutions. Our results show notably that neither the ill-posedness nor the proportion α play any role in the bounds whenever α ∈ [0, 1), and that a particular inconsistency zone appears for some values of the parameters. Moreover, we introduce an additional boundedness condition on f and we show that the inconsistency zone then disappears.AMS 2000 subject classifications: 62G05, 62G20.

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Section: Are the Lower Bounds Sharp?mentioning

confidence: 85%

Lower bounds in the convolution structure density model

Lepski¹,

Willer²

2017

Bernoulli

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“…Let us also remark that there is no lower bound result in Masry (1993). The most general results in the deconvolution model were obtained in Comte and Lacour (2013) and Rebelles (2016) and in Section 4 we will compare in detail our results with those obtained in these papers.…”

mentioning

confidence: 72%

Oracle inequalities and adaptive estimation in the convolution structure density model

Lepski¹,

Willer²

2019

Ann. Statist.

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We study the problem of nonparametric estimation under Lp-loss, p ∈ [1, ∞), in the framework of the convolution structure density model on R d . This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. The original pointwise selection rule from a family of "kernel-type" estimators is proposed. For the selected estimator, we prove an Lp-norm oracle inequality and several of its consequences. Next, the problem of adaptive minimax estimation under Lp-loss over the scale of anisotropic Nikol'skii classes is addressed. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. We prove that the proposed selection rule leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator.

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“…Results have also been established for multivariate anisotropic densities (see e.g. [16] for L 2 -loss functions or [50] for L p -loss functions, p ∈ [1, ∞]). Deconvolution with unknown error distribution has also been studied (see e.g.…”

Section: Statistical Settingmentioning

confidence: 99%

Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

Duval¹,

Kappus²

2019

Electron. J. Statist.

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We introduce a new procedure to select the optimal cutoff parameter for Fourier density estimators that leads to adaptive rate optimal estimators, up to a logarithmic factor. This adaptive procedure applies for different inverse problems. We illustrate it on two classical examples: deconvolution and decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of n increments of length ∆ > 0. For this latter example, we first build an estimator for which we provide an upper bound for its L 2 -risk that is valid simultaneously for sampling rates ∆ that can vanish, ∆ := ∆ n → 0, can be fixed, ∆ n → ∆ 0 > 0 or can get large, ∆ n → ∞ slowly. This last result is new and presents interest on its own. Then, we show that the adaptive procedure we present leads to an adaptive and rate optimal (up to a logarithmic factor) estimator of the jump density.

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Structural adaptive deconvolution under $${\mathbb{L}_p}$$ -losses

Cited by 13 publications

References 41 publications

Lower bounds in the convolution structure density model

Lower bounds in the convolution structure density model

Oracle inequalities and adaptive estimation in the convolution structure density model

Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

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